(*  Title:      HOL/Bali/Basis.thy

    Author:     David von Oheimb

*)

header {* Definitions extending HOL as logical basis of Bali *}

theory Basis

imports Main "~~/src/HOL/Library/Old_Recdef"

begin

section "misc"

ML {* fun strip_tac i = REPEAT (resolve_tac [impI, allI] i) *}

declare split_if_asm  [split] option.split [split] option.split_asm [split]

declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}

declare if_weak_cong [cong del] option.weak_case_cong [cong del]

declare length_Suc_conv [iff]

lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"

  by auto

lemma subset_insertD: "A \<subseteq> insert x B \<Longrightarrow> A \<subseteq> B \<and> x \<notin> A \<or> (\<exists>B'. A = insert x B' \<and> B' \<subseteq> B)"

  apply (case_tac "x \<in> A")

   apply (rule disjI2)

   apply (rule_tac x = "A - {x}" in exI)

   apply fast+

  done

abbreviation nat3 :: nat  ("3") where "3 \<equiv> Suc 2"

abbreviation nat4 :: nat  ("4") where "4 \<equiv> Suc 3"

(* irrefl_tranclI in Transitive_Closure.thy is more general *)

lemma irrefl_tranclI': "r\<inverse> \<inter> r\<^sup>+ = {} \<Longrightarrow> \<forall>x. (x, x) \<notin> r\<^sup>+"

  by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma trancl_rtrancl_trancl: "\<lbrakk>(x, y) \<in> r\<^sup>+; (y, z) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (x, z) \<in> r\<^sup>+"

  by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)

lemma rtrancl_into_trancl3: "\<lbrakk>(a, b) \<in> r\<^sup>*; a \<noteq> b\<rbrakk> \<Longrightarrow> (a, b) \<in> r\<^sup>+"

  apply (drule rtranclD)

  apply auto

  done

lemma rtrancl_into_rtrancl2: "\<lbrakk>(a, b) \<in>  r; (b, c) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (a, c) \<in> r\<^sup>*"

  by (auto intro: rtrancl_trans)

lemma triangle_lemma:

  assumes unique: "\<And>a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b = c"

    and ax: "(a,x)\<in>r\<^sup>*" and ay: "(a,y)\<in>r\<^sup>*"

  shows "(x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"

  using ax ay

proof (induct rule: converse_rtrancl_induct)

  assume "(x,y)\<in>r\<^sup>*"

  then show ?thesis by blast

next

  fix a v

  assume a_v_r: "(a, v) \<in> r"

    and v_x_rt: "(v, x) \<in> r\<^sup>*"

    and a_y_rt: "(a, y) \<in> r\<^sup>*"

    and hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"

  from a_y_rt show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"

  proof (cases rule: converse_rtranclE)

    assume "a = y"

    with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"

      by (auto intro: rtrancl_trans)

    then show ?thesis by blast

  next

    fix w

    assume a_w_r: "(a, w) \<in> r"

      and w_y_rt: "(w, y) \<in> r\<^sup>*"

    from a_v_r a_w_r unique have "v=w" by auto

    with w_y_rt hyp show ?thesis by blast

  qed

qed

lemma rtrancl_cases:

  assumes "(a,b)\<in>r\<^sup>*"

  obtains (Refl) "a = b"

    | (Trancl) "(a,b)\<in>r\<^sup>+"

  apply (rule rtranclE [OF assms])

   apply (auto dest: rtrancl_into_trancl1)

  done

lemma Ball_weaken: "\<lbrakk>Ball s P; \<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"

  by auto

lemma finite_SetCompr2:

  "finite (Collect P) \<Longrightarrow> \<forall>y. P y \<longrightarrow> finite (range (f y)) \<Longrightarrow>

    finite {f y x |x y. P y}"

  apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (\<lambda>y. range (f y))")

   prefer 2 apply fast

  apply (erule ssubst)

  apply (erule finite_UN_I)

  apply fast

  done

lemma list_all2_trans: "\<forall>a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>

    \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"

  apply (induct_tac xs1)

   apply simp

  apply (rule allI)

  apply (induct_tac xs2)

   apply simp

  apply (rule allI)

  apply (induct_tac xs3)

   apply auto

  done

section "pairs"

lemma surjective_pairing5:

  "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),

    snd (snd (snd (snd p))))"

  by auto

lemma fst_splitE [elim!]:

  assumes "fst s' = x'"

  obtains x s where "s' = (x,s)" and "x = x'"

  using assms by (cases s') auto

lemma fst_in_set_lemma: "(x, y) : set l \<Longrightarrow> x : fst ` set l"

  by (induct l) auto

section "quantifiers"

lemma All_Ex_refl_eq2 [simp]: "(\<forall>x. (\<exists>b. x = f b \<and> Q b) \<longrightarrow> P x) = (\<forall>b. Q b \<longrightarrow> P (f b))"

  by auto

lemma ex_ex_miniscope1 [simp]: "(\<exists>w v. P w v \<and> Q v) = (\<exists>v. (\<exists>w. P w v) \<and> Q v)"

  by auto

lemma ex_miniscope2 [simp]: "(\<exists>v. P v \<and> Q \<and> R v) = (Q \<and> (\<exists>v. P v \<and> R v))"

  by auto

lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"

  by auto

lemma All_Ex_refl_eq1 [simp]: "(\<forall>x. (\<exists>b. x = f b) \<longrightarrow> P x) = (\<forall>b. P (f b))"

  by auto

section "sums"

hide_const In0 In1

notation sum_case  (infixr "'(+')"80)

primrec the_Inl :: "'a + 'b \<Rightarrow> 'a"

  where "the_Inl (Inl a) = a"

primrec the_Inr :: "'a + 'b \<Rightarrow> 'b"

  where "the_Inr (Inr b) = b"

datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c

primrec the_In1 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"

  where "the_In1 (In1 a) = a"

primrec the_In2 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"

  where "the_In2 (In2 b) = b"

primrec the_In3 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"

  where "the_In3 (In3 c) = c"

abbreviation In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"

  where "In1l e \<equiv> In1 (Inl e)"

abbreviation In1r :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"

  where "In1r c \<equiv> In1 (Inr c)"

abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"

  where "the_In1l \<equiv> the_Inl \<circ> the_In1"

abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"

  where "the_In1r \<equiv> the_Inr \<circ> the_In1"

ML {*

fun sum3_instantiate ctxt thm = map (fn s =>

  simplify (simpset_of ctxt delsimps [@{thm not_None_eq}])

    (read_instantiate ctxt [(("t", 0), "In" ^ s ^ " ?x")] thm)) ["1l","2","3","1r"]

*}

(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)

section "quantifiers for option type"

syntax

  "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3! _:_:/ _)" [0,0,10] 10)

  "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3? _:_:/ _)" [0,0,10] 10)

syntax (symbols)

  "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)

  "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)

translations

  "\<forall>x\<in>A: P" \<rightleftharpoons> "\<forall>x\<in>CONST Option.set A. P"

  "\<exists>x\<in>A: P" \<rightleftharpoons> "\<exists>x\<in>CONST Option.set A. P"

section "Special map update"

text{* Deemed too special for theory Map. *}

definition chg_map :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"

  where "chg_map f a m = (case m a of None \<Rightarrow> m | Some b \<Rightarrow> m(a\<mapsto>f b))"

lemma chg_map_new[simp]: "m a = None \<Longrightarrow> chg_map f a m = m"

  unfolding chg_map_def by auto

lemma chg_map_upd[simp]: "m a = Some b \<Longrightarrow> chg_map f a m = m(a\<mapsto>f b)"

  unfolding chg_map_def by auto

lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"

  by (auto simp: chg_map_def)

section "unique association lists"

definition unique :: "('a \<times> 'b) list \<Rightarrow> bool"

  where "unique = distinct \<circ> map fst"

lemma uniqueD: "unique l \<Longrightarrow> (x, y) \<in> set l \<Longrightarrow> (x', y') \<in> set l \<Longrightarrow> x = x' \<Longrightarrow> y = y'"

  unfolding unique_def o_def

  by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_Nil [simp]: "unique []"

  by (simp add: unique_def)

lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l \<and> (\<forall>y. (x,y) \<notin> set l))"

  by (auto simp: unique_def dest: fst_in_set_lemma)

lemma unique_ConsD: "unique (x#xs) \<Longrightarrow> unique xs"

  by (simp add: unique_def)

lemma unique_single [simp]: "\<And>p. unique [p]"

  by simp

lemma unique_append [rule_format (no_asm)]: "unique l' \<Longrightarrow> unique l \<Longrightarrow>

    (\<forall>(x,y)\<in>set l. \<forall>(x',y')\<in>set l'. x' \<noteq> x) \<longrightarrow> unique (l @ l')"

  by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_map_inj: "unique l \<Longrightarrow> inj f \<Longrightarrow> unique (map (\<lambda>(k,x). (f k, g k x)) l)"

  by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)

lemma map_of_SomeI: "unique l \<Longrightarrow> (k, x) : set l \<Longrightarrow> map_of l k = Some x"

  by (induct l) auto

section "list patterns"

definition lsplit :: "[['a, 'a list] \<Rightarrow> 'b, 'a list] \<Rightarrow> 'b"

  where "lsplit = (\<lambda>f l. f (hd l) (tl l))"

text {* list patterns -- extends pre-defined type "pttrn" used in abstractions *}

syntax

  "_lpttrn" :: "[pttrn, pttrn] \<Rightarrow> pttrn"    ("_#/_" [901,900] 900)

translations

  "\<lambda>y # x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>y x # xs. b)"

  "\<lambda>x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>x xs. b)"

lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"

  by (simp add: lsplit_def)

lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"

  by (simp add: lsplit_def)

end